(where c is a constant of integration) |
A) \[{{y}^{2}}+2{{x}^{3}}+c{{x}^{2}}=0\]
B) \[{{y}^{2}}+2{{x}^{2}}+c{{x}^{3}}=0\]
C) \[{{y}^{2}}2{{x}^{3}}+c{{x}^{2}}=0\]
D) \[{{y}^{2}}2{{x}^{2}}+c{{x}^{3}}=0\]
Correct Answer: A
Solution :
\[xy\frac{dy}{dx}-{{y}^{2}}+{{x}^{3}}=0\] put\[{{y}^{2}}=k\Rightarrow y\frac{dy}{dx}=\frac{1}{2}\frac{dk}{dx}\] \[\therefore \]given differential equation becomes \[\frac{dk}{dx}+k\left( -\frac{2}{x} \right)=-2{{x}^{2}}\] I.F.\[={{e}^{\int_{{}}^{{}}{-\frac{2}{x}dx}}}=\frac{1}{{{x}^{2}}}\] \[\therefore \]solution is \[k.\frac{1}{{{x}^{2}}}=\int_{{}}^{{}}{-2{{x}^{2}}.\frac{1}{{{x}^{2}}}dx+\lambda }\] \[{{y}^{2}}+2{{x}^{3}}=\lambda {{x}^{2}}\] take \[\lambda =-c\](integration constant)You need to login to perform this action.
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